Probability & Statistics PYQ — Page 9

In a group of $400$ people, $160$ are smokers and non-vegetarian; $100$ are smokers and vegetarian and the remaining $140$ are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35%,20%$ and $10%$ respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is :

An online exam is attempted by $50$ candidates out of which $20$ are boys. The average marks obtained by boys is $12$ with a variance $2.$ The variance of marks obtained by $30$ girls is also $2.$ The average marks of all $50$ candidates is $15.$ If $\mu$ is the average marks of girls and ${\sigma }^{2}$ is the variance of marks of $50$ candidates, then $\mu +{\sigma }^{2}$ is equal to

Let there be three independent events ${E}_{1},{E}_{2}$ and ${E}_{3}.$ The probability that only ${E}_{1}$ occurs is $\alpha$ only ${E}_{2}$ occurs is $\beta$ and only ${E}_{3}$ occurs is $\gamma .$ Let $p''$ denote the probability of none of events occurs that satisfies the equations $(\alpha -2\beta )p=\alpha \beta$ and $(\beta -3\gamma )p=2\beta \gamma .$ All the given probabilities are assumed to lie in the interval $(0,1).$ Then, $\frac{\text{ Probability of occurrence of }{E}_{1}}{\text{ Probability of occurrence of }{E}_{3}}$ is equal to ________.

If the mean and variance of the following data: $6,10,7,13,a,12,b,12$ are $9$ and $\frac{37}{4}$ respectively, then ${(a-b)}^{2}$ is equal to:

Let in a series of $2n$ observations, half of them are equal to $a$ and remaining half are equal to $-a$. Also by adding a constant $b$ in each of these observations, the mean and standard deviation of new set become $5$ and $20$, respectively. Then the value of ${a}^{2}+{b}^{2}$ is equal to :

The mean age of $25$ teachers in a school is $40$ years. A teacher retires at the age of $60$ years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is $39$ years, then the age (in years) of the newly appointed teacher is

The probability distribution of random variable $X$ is given by: <table class="pyq-table"><tbody><tr><td>$X$</td><td>$1$</td><td>$2$</td><td>$3$</td><td>$4$</td><td>$5$</td></tr><tr><td>$P(X)$</td><td>$K$</td><td>$2K$</td><td>$2K$</td><td>$3K$</td><td>$K$</td></tr></tbody></table>Let $p=P(1<X<4\mid X<3)$. If $5p=\lambda K$, then $\lambda$ is equal to

Words with or without meaning are to be formed using all the letters of the word $\mathrm{EXAMINATION}$. The probability that the letter $M$ appears at the fourth position in any such word is:

The mean of $10$ numbers $7\times 8,10\times 10,13\times 12,16\times 14,\ldots$ is

Let $A$ denote the event that a $6$-digit integer formed by $0,1,2,3,4,5,6$ without repetitions, be divisible by $3$ . Then probability of event $A$ is equal to :

If the mean and variance of six observations $7,10,11,15,a,b$ are $10$ and $\frac{20}{3},$ respectively, then the value of $|a-b|$ is equal to:

A fair die is tossed until six is obtained on it. Let $X$ be the number of required tosses, then the conditional probability $P(X\geqslant 5\mid X>2)$ is :

Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is : 

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $2\times 2$ matrices. The probability that such formed matrices have all different entries and are non-singular, is:

Two dices are rolled. If both dices have six faces numbered $1,2,3,5,7$ and $11,$ then the probability that the sum of the numbers on the top faces is less than or equal to $8$ is:

A seven digit number is formed using digits $3,3,4,4,4,5,5$. The probability, that number so formed is divisible by $2$, is

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number $2$ times is equal to the probability of getting an even number $3$ times, then the probability of getting an odd number for odd number of times is:

The probability that two randomly selected subsets of the set ${1,2,3,4,5}$ have exactly two elements in their intersection, is:

When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}$ and the probability that the missile hits the target, given that it is not intercepted, is $\frac{3}{4}.$ If three missiles are fired independently from the ship, then the probability that all three hit the target, is:

The mean of $6$ distinct observations is $6.5$ and their variance is $10.25.$ If $4$ out of $6$ observations are $2,4,5$ and $7,$ then the remaining two observations are:

The first of the two samples in a group has $100$ items with mean $15$ and standard deviation $3.$ If the whole group has $250$ items with mean $15.6$ and standard deviation $\sqrt{13.44},$ then the standard deviation of the second sample is:

Let $A$ be a set of all $4$ -digit natural numbers whose exactly one digit is $7$. Then the probability that a randomly chosen element of $A$ leaves remainder $2$ when divided by $5$ is:

The mean and variance of $7$ observations are $8$ and $16$ respetively. If two observations are $6$ and $8,$ then the variance of the remaining $5$ observations is :

Let ${B}_{i}(i=1,2,3)$ be three independent events in a sample space. The probability that only ${B}_{1}$ occur is $\alpha ,$ only ${B}_{2}$ occurs is $\beta$ and only ${B}_{3}$ occurs is $\gamma .$ Let $p$ be the probability that none of the events ${B}_{i}$ occurs and these $4$ probabilities satisfy the equations $(\alpha -2\beta )p=\alpha \beta$ and $(\beta -3\gamma )p=2\beta \gamma$ (All the probabilities are assumed to lie in the interval $(0,1))$ Then $\frac{P({B}_{1})}{P({B}_{3})}$ is equal to______.